Abstract: Let R be a ring and let M be an R-module with S=\rm{End}_R(M). Consider the preradical {\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu} for the category of right R-modules Mod-R introduced by Y. Talebi and N. Vanaja in 2002 and defined by {\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}(M) = \bigcap \{U\leq M\colon M/U is small in its injective hull\}. The module M is called quasi-t-dual Baer if \sum_{\varphi \in \mathfrak{I}} \varphi({{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M)) is a direct summand of M for every two-sided ideal \mathfrak{I} of S, where {{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M) = {{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}} ({{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}(M)). In this paper, we show that M is quasi-t-dual Baer if and only if {{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M) is a direct summand of M and {\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}^2(M) is a quasi-dual Baer module. It is also shown that any direct summand of a quasi-t-dual Baer module inherits the property. The last part of the paper is devoted to the comparison of the notions of quasi-dual Baer modules and quasi-t-dual Baer modules. Also, right quasi-t-dual Baer rings are investigated.
Keywords: fully invariant submodule; quasi-dual Baer module; quasi-dual Baer ring; quasi-t-dual Baer module; quasi-t-dual Baer ring
DOI: DOI 10.14712/1213-7243.2024.008
AMS Subject Classification: 16D10 16D80